Images of A- and K-orbits under the parabolic Cayley transform¶

Exercise I.10.11 [1] Verify that:

$A$-orbits in transforms $P_e$ and $P_h$ are segments of parabolas with focal length $\frac{1}{4}$ and passing through $(0,-1)$. Their p-foci (i.e. vertices) belong to two parabolas $v=(-u^2-1)$ and $v=(u^2-1)$ respectively, which are the boundaries of parabolic circles in $P_h$ and $P_e$ (note the swap!).
$K$-orbits in transform $P_e$ are parabolas with focal length less than $\frac{1}{4}$. In transform $P_h$, they are parabolas where the reciprocal of the focal length is larger than $-4$.

Solution. We verify properties of A orbits under the Cayley transform. The next cycle represent a generic A-orbit.

from init_cycle import *
Ca=cayley(cycle2D(0,[1,t],0,e),0,sign1)

            Python wrappers for MoibInv Library
     ---------------------------------------------
Please cite this software as
V.V. Kisil, MoebInv: C++ libraries for manipulations in non-Euclidean geometry, SoftwareX, 11(2020),100385. doi:10.1016/j.softx.2019.100385.
     ---------------------------------------------

Using vector formalism and idx

The properties of the parabolic Cayley transform of Ca are:

display(Latex("Focal length of A-orbit: $%s$" % Ca.focal_length()))
print("A-orbit passes (0,-1): %s" % Ca.val([0,-1]).subs({sign : 0}).is_zero())
Pf=Ca.focus(diag_matrix([-1,0]))
display(Latex("p-focus of A-orbit belong to the parabola $v=\\breve{\\sigma} u^2-1$: %s" %\
(sign1*pow(Pf[0],2)-Pf[1]-1).normal().is_zero()))

A-orbit passes (0,-1): True

A generic K-orbit and its properties are:

Ck=cayley(cycle2D(1,[0,t],1,e),0,sign1)
display(Latex("Focal length of K-orbit in P_e is less than $1/4$ by: $%s$" % (numeric(1,4)-Ck.focal_length()).subs({sign1 : -1}).normal()))
display(Latex("Reciprocal of focal length of K-orbit in P_h is bigger than $-4$ by: $%s$" % (4+1/Ck.focal_length()).subs({sign1 : 1}).normal()))

This notebook is a part of the MoebInv notebooks project [2] .

References¶

Vladimir V. Kisil. Geometry of Möbius Transformations: Elliptic, Parabolic and Hyperbolic Actions of $SL_2(\mathbb{R})$. Imperial College Press, London, 2012. Includes a live DVD.
Vladimir V. Kisil, MoebInv notebooks, 2019.

Back to Folder